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106 V NORMED SPACES

(a) Show that if/satisfies condition (1) and is continuous at the point 0, it is con-
tinuous in a neighborhood of 0 and is linear in a neighborhood of 0 (cf. Problem 1).

(b) Let g be a mapping of E into F; in order that g be continuous at the point 0 and
linear in a neighborhood of 0, a necessary and sufficient condition is that for any
convergent series (x_n) in E, the partial sums of the series (g(x_n) be bounded in F.
(To prove sufficiency, first observe that one must have #(0) = 0; if, for every n, there
exist three elements u»,v_n,w_n of E such that \\u_n\\ < 2~", ||oj| < 2~", Hw.ll ^2-",
u_n + v_n 4- M>_B = 0 and#(w_n) + g(v_n) +g(w_n) ^ 0, form a series (x_n) violating the assump-
tion. If there are no such sequences u_n, v_n, w_n, g verifies condition (1); show that it is
necessarily continuous at 0.)

6. EQUIVALENT NORMS

Let E be a vector space (over the real or the complex field), \\x\\i and
||A:||₂ two norms on E; we say that \\x\\i is finer than ||#||₂ if the topology
defined by \\x\\i is finer than the topology defined by ||jc||₂ (see Section 3.12);
if we note E_l (resp. E₂) the normed space determined by \\x\\i (resp. ||jc||₂),
this means that the identity mapping x -> x of E! into E₂ is continuous,
hence, by (5.5.1), that condition is equivalent to the existence of a number
a>0 such that ||x||₂ < a- \\x\\_l9 We say that the two norms \\x\\_l9 \\x\\₂are equivalent if they define the same topology on E. The preceding remark
yields at once:

(5.6.1) In order that the two norms \\x\\i, \\x\\ ₂ona vector space E be equivalent
a necessary and sufficient condition is that there exist two constants a > 0,
i>0, such that

«Wi<IMl2<6|l*lli

for any xeE.

The corresponding distances are then uniformly equivalent (Section 3.14).

For instance, on the product E_r x E₂ of two normed spaces, the norms
sup(KIUM), \M\ + \\x₂\\, (\\xtf + \\x₂\\²)¹'² are equivalent. On the
space E = ^_R(I), the norm H/ld defined in (5.1.4) is not equivalent to the
norm \\f\\_n = sup|/(f)| (see Section 5.1, Problem 1).

tel

7. SPACES OF CONTINUOUS MULTILINEAR MAPPINGS

Let E, F, be two normed spaces; the set jSf(E;F) of all continuous
linear mappings of E into F is a vector space, as follows from (5.1.5), (3.20.4),
and (3.11.5).



	106 V NORMED SPACES (a) Show that if/satisfies condition (1) and is continuous at the point 0, it is con- tinuous in a neighborhood of 0 and is linear in a neighborhood of 0 (cf. Problem 1). (b) Let g be a mapping of E into F; in order that g be continuous at the point 0 and linear in a neighborhood of 0, a necessary and sufficient condition is that for any convergent series (x_n) in E, the partial sums of the series (g(x_n) be bounded in F. (To prove sufficiency, first observe that one must have #(0) = 0; if, for every n, there exist three elements u»,v_n,w_n of E such that \\u_n\\ < 2~", \|\|oj\| < 2~", Hw.ll ^2-", u_n + v_n 4- M>_B = 0 and#(w_n) + g(v_n) +g(w_n) ^ 0, form a series (x_n) violating the assump- tion. If there are no such sequences u_n, v_n, w_n, g verifies condition (1); show that it is necessarily continuous at 0.)

	6. EQUIVALENT NORMS Let E be a vector space (over the real or the complex field), \\x\\i and \|\|A:\|\|₂ two norms on E; we say that \\x\\i is finer than \|\|#\|\|₂ if the topology defined by \\x\\i is finer than the topology defined by \|\|jc\|\|₂ (see Section 3.12); if we note E_l (resp. E₂) the normed space determined by \\x\\i (resp. \|\|jc\|\|₂), this means that the identity mapping x -> x of E! into E₂ is continuous, hence, by (5.5.1), that condition is equivalent to the existence of a number a>0 such that \|\|x\|\|₂ < a- \\x\\_l9 We say that the two norms \\x\\_l9 \\x\\₂are equivalent if they define the same topology on E. The preceding remark yields at once:

	(5.6.1) In order that the two norms \\x\\i, \\x\\ ₂ona vector space E be equivalent a necessary and sufficient condition is that there exist two constants a > 0, i>0, such that «Wi<IMl2<6\|llli for any xeE.* The corresponding distances are then uniformly equivalent (Section 3.14). For instance, on the product E_r x E₂ of two normed spaces, the norms sup(KIUM), \M\ + \\x₂\\, (\\xtf + \\x₂\\²)¹'² are equivalent. On the space E = ^_R(I), the norm H/ld defined in (5.1.4) is not equivalent to the norm \\f\\_n = sup\|/(f)\| (see Section 5.1, Problem 1). *tel*

	7. SPACES OF CONTINUOUS MULTILINEAR MAPPINGS Let E, F, be two normed spaces; the set jSf(E;F) of all continuous linear mappings of E into F is a vector space, as follows from (5.1.5), (3.20.4), and (3.11.5).